LMFDB - L-function 2-1900-19.18-c2-0-12

Dirichlet series

L(s) = 1

− 2.03i·3^-s − 4.27·7^-s + 4.86·9^-s + 5.47·11^-s − 0.634i·13^-s + 1.69·17^-s + (−6.46 + 17.8i)19^-s + 8.70i·21^-s − 25.9·23^-s − 28.1i·27^-s + 48.8i·29^-s + 4.47i·31^-s − 11.1i·33^-s + 6.12i·37^-s − 1.28·39^-s + ⋯

L(s) = 1

− 0.677i·3^-s − 0.611·7^-s + 0.540·9^-s + 0.497·11^-s − 0.0487i·13^-s + 0.0995·17^-s + (−0.340 + 0.940i)19^-s + 0.414i·21^-s − 1.12·23^-s − 1.04i·27^-s + 1.68i·29^-s + 0.144i·31^-s − 0.337i·33^-s + 0.165i·37^-s − 0.0330·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.340 - 0.940i$
Analytic conductor:	$51.7712$
Root analytic conductor:	$7.19522$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (1101, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1),\ 0.340 - 0.940i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.210152537$
$L(\frac12)$	$\approx$	$1.210152537$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
	19	$ 1 + (6.46 - 17.8i)T $
good	3	$ 1 + 2.03iT - 9T^{2} $
	7	$ 1 + 4.27T + 49T^{2} $
	11	$ 1 - 5.47T + 121T^{2} $
	13	$ 1 + 0.634iT - 169T^{2} $
	17	$ 1 - 1.69T + 289T^{2} $
	23	$ 1 + 25.9T + 529T^{2} $
	29	$ 1 - 48.8iT - 841T^{2} $
	31	$ 1 - 4.47iT - 961T^{2} $
	37	$ 1 - 6.12iT - 1.36e3T^{2} $
	41	$ 1 - 16.9iT - 1.68e3T^{2} $
	43	$ 1 + 0.690T + 1.84e3T^{2} $
	47	$ 1 + 23.0T + 2.20e3T^{2} $
	53	$ 1 + 54.3iT - 2.80e3T^{2} $
	59	$ 1 + 0.251iT - 3.48e3T^{2} $
	61	$ 1 - 39.5T + 3.72e3T^{2} $
	67	$ 1 - 96.3iT - 4.48e3T^{2} $
	71	$ 1 + 16.0iT - 5.04e3T^{2} $
	73	$ 1 + 70.0T + 5.32e3T^{2} $
	79	$ 1 + 74.6iT - 6.24e3T^{2} $
	83	$ 1 + 1.29T + 6.88e3T^{2} $
	89	$ 1 - 141. iT - 7.92e3T^{2} $
	97	$ 1 - 125. iT - 9.40e3T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.177297724019114642510276762457, −8.284999066585267747239092098539, −7.59568320064634338761427615541, −6.69772439499376673459024705209, −6.29439070218387009213504473159, −5.23507662343874668482054331069, −4.11244887968404643494353119499, −3.34311507955629835524224572165, −2.05079336090139235852245843126, −1.16196528195770947900368980586, 0.32663671763078205653472957535, 1.85260680932020921224231725323, 3.04677363009197300085366985627, 4.05449245533850116561471149986, 4.53312663736291716218224408424, 5.71146494089040899626149243032, 6.46746360071226113790279952257, 7.24984156255320277224636151613, 8.165149099154444673644031538157, 9.096567750558406043607903503449

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.210152537\)
\(L(\frac12)\)	\(\approx\)	\(1.210152537\)
\(L(2)\)		not available
\(L(1)\)		not available

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